Many areas of active research within the broad field of number theory relate to properties of polynomials, and this volume displays the most recent and most interesting work on this theme. The 2006 Number Theory and Polynomials workshop in Bristol drew together international researchers with a variety of number-theoretic interests, and the book's contents reflect the quality of the meeting. Topics covered include recent work on the Schur-Siegel-Smyth trace problem, Mahler measure and its generalisations, the merit factor problem, Barker sequences, K3-surfaces, self-inversive polynomials, Newman's inequality, algorithms for sparse polynomials, the integer transfinite diameter, divisors of polynomials, non-linear recurrence sequences, polynomial ergodic averages, and the Hansen-Mullen primitivity conjecture. With surveys and expository articles presenting the latest research, this volume is essential for graduates and researchers looking for a snapshot of current progress in polynomials and number theory.
Author(s): James McKee, Chris Smyth
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 2008
Language: English
Pages: 364
Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Contents ......Page 6
Preface ......Page 8
Index of authors ......Page 9
List of participants ......Page 12
Conference photograph, with key ......Page 0
The trace problem for totally positive algebraic integers with an appendix by Jean-Pierre Serre ......Page 16
Mahler's measure: from Number Theory to Geometry ......Page 35
Explicit calculation of elliptic fibrations of K^3-surfaces and their Belyi-maps. ......Page 48
The merit factor problem ......Page 67
Barker sequences and flat polynomials. ......Page 86
The Hansen-Mullen primitivity conjecture: completion of proof. ......Page 104
An inequality for the multiplicity of the roots of a polynomial ......Page 136
Newman's inequality for increasing exponential sums ......Page 142
On primitive divisors of n^2 + b. ......Page 157
Irreducibility and greatest common divisor algorithms for sparse polynomials. ......Page 170
Consequences of the continuity of the monic integer transfinite diameter. ......Page 192
Nonlinear recurrence sequences and Laurent polynomials. ......Page 203
Conjugate algebraic numbers on conics: a survey ......Page 226
On polynomial ergodic averages and square functions ......Page 256
Polynomial inequalities, Mahler's measure, and multipliers ......Page 270
Integer transfinite diameter and computation of polynomials. ......Page 292
Smooth divisors of polynomials. ......Page 301
Self-inversive polynomials with all zeros on the unit circle. ......Page 327
The Mahler measure of algebraic numbers: a survey. ......Page 337
